Homotopy continuation for the spectra of persistent Laplacians

Figure 1.  0-simplex, 1-simplex, 2-simplex and 3-simplex

Figure 2.  Rips complexes corresponding to different $ r $ values

Figure 3.  Pentagon, Heptagon, Octagon and Nonagon

Figure 4.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ against the radius $ \alpha $ for the pentagon. The persistent Betti numbers are shown by the blue line. The smallest nonzero eigenvalues calculated by HERMES and Bertini are shown by a red line and red circles, respectively and one can see that they almost coincide. The half edge length is approximately $ \sin(\pi/5) \approx 0.58 $

Figure 5.  (a) A cube with edge length 1. (b) A regular octahedron with edge length $ \sqrt{2} $. (c) A regular tetrahedron with edge length $ \sqrt{3} $. (d) A regular pyramid with square edge length $ \sqrt{2} $ and height 2. (e) A triangular prism with edge length $ \sqrt{3}$

Figure 6.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for the cube. The length of its face diagonal is $ \sqrt{2} \approx 1.4 $ and the length of its main diagonal is $ \sqrt{3} \approx 1.7$

Figure 7.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for the octahedron. Its edge length is $ \sqrt{2} $. The circumradius of any face is equal to $ \sqrt{2}/\sqrt{3} \approx 0.8 $. The circumradius of the octahedron itself is equal to $ 1 $

Figure 8.  Some aromatic molecules

Figure 9.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for the benzene. The half length of its edge is approximately 0.7Å, and its radius is approximately 1.4Å

Figure 10.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for the heptagon

Figure 11.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for the octagon

Figure 12.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for the nonagon

Figure 13.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for the regular tetrahedron with edge length $ \sqrt{3} $. The centroid to vertex distance is $ 3/\sqrt{8} \approx 1.06 $

Figure 14.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for a triangular prism. Its base is a regular triangle with edge length $ \sqrt{3} $. Its side faces are squares. Important distances are $ \sqrt{3}/2, 1, \sqrt{6}/2 \approx 1.22 $ and $ \sqrt{7}/2 \approx 1.32 $

Figure 15.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for a regular pyramid. Its base is a square with edge length $ \sqrt{2} $ and its height is 2. Important distances are $ \sqrt{2}/2, 1, \sqrt{5}/2 \approx 1.12, 5\sqrt{2}/6 \approx 1.18 $ and $ 5/4 $

Figure 16.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for the naphthalene

Figure 17.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for the anthracene

Figure 18.  The illustration of the persistent Betti numbers $ \beta_{q}^{\alpha, 0} $ and the smallest nonzero eigenvalues of persistent Laplacians $ \lambda_{q}^{\alpha, 0} $ for the pyrene